First slide

Compound angles

Question

If $\frac{\tan x}{2} = \frac{\tan y}{3} = \frac{\tan z}{5}$ and $x + y + z = π$ then the value of $\tan^{2} x + \tan^{2} y + \tan^{2} z$ is

Moderate

A

$\frac{38}{3}$
B

38
C

114
D

none of these

Solution

$x + y + z = π \Rightarrow \tan x + \tan y + \tan z =$ $\tan x \tan y \tan z$
$\Rightarrow 2 k + 3 k + 5 k = 2 k \times 3 k \times 5 k \Rightarrow k^{2} = 1 / 3$
$\tan^{2} x + \tan^{2} y + \tan^{2} z = \frac{1}{3} (4 + 9 + 25) = \frac{38}{3}$ .

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